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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.8, Problem 11E

(a)

To determine

**To find:** The value of

Expert Solution

The value of

**Given:**

The company makes square wafers of silicon of area of the form **.**

**Calculation:**

Differentiate the area of the square wafer respect to *x.*

Substitute 15 for

Therefore, the rate at which the area will increase with respect to change in its side length, if the side length is 15 mm will be

(b)

To determine

**To Show:** The rate of change of the area of a square with respect to the change in its side length is equal to half its perimeter.

Expert Solution

Draw the square with side length, *x* as shown in Figure (1).

The perimeter of the square is

From the figure 1, the change in length

The first derivative of

Thus, the change in area of the square is approximately half of its perimeter which is half of the 4 sides times.

The expression for the shaded area (change in area) as below.

The change in length

The term

Thus the rate of change of the area of a square with respect to the change in its side length is equal to half its perimeter is